Topological spaces that are homeomorphic but not PL-homeomorphic I am trying to understand basic concepts in piecewise-linear topology. It seems the most basic concept is "piecewise-linear homeomorphism". What are simple examples of topological spaces manifolds, that are homeomorphic but not PL-homeomorphic?
[The nearest answer I found was here: the Milnor sphere and the standard sphere are homeomorphic, and PL-homeomorphic, but they are not diffeomorphic.]
 A: I will ignore the request for a "simple" example: As far as I know, all the known examples require quite a bit of hard work.
Theorem. For every $n\ge 5$ there exists a triangulated manifold $(M,\tau)$ homeomorphic to the standard $n$-dimensional torus $T^n$ but not PL homeomorphic to it.
Proof. The reference is essentially page 3 of "The Hauptvermutung book." By reading page 3,  you see that there exists $(M,\tau)$ and a homeomorphism
$$
f: (M,\tau)\to T^n
$$
not homotopic to a PL homeomorphism. (The proof is quite hard.) Suppose that
$$
g: (M,\tau)\to T^n
$$
is another homeomorphism. Then the composition $f\circ g^{-1}: T^n\to T^n$ is a self-homeomorphism. Each self-homotopy-equivalence of $T^n$ is homotopic to an affine map $A: T^n\to T^n$ (i.e. a map which lifts to an affine self-map the universal covering space of the standard torus, which is the Euclidean space $E^n$): This is a nice exercise in application of Whitehead'd theorem on CW complexes with contractible universal covers. An affine self-map of the torus is PL, of course. Thus, if $g$ were to be a PL homeomorphism $(M,\tau)\to T^n$, then $f$ would be homotopic to the PL homeomorphism $A\circ g$, which is a contradiction. Thus, there are no PL homeomorphisms $(M,\tau)\to T^n$.
A: There are PL-manifolds which are homeomorphic but not PL-homeomorphic. This is a highly non-trivial result.
See Wikipedia. A more detailed exposition can be found in

Ranicki, A. A., et al. "The hauptvermutung book." Collection of papers by Casson, Sullivan, Armstrong, Cooke, Rourke and Ranicki, K-Monographs in Mathematics 1 (1996).

In

Kirby, R. C., & Siebenmann, L. C. (1969). On the triangulation of manifolds and the Hauptvermutung. Bulletin of the American Mathematical Society, 75(4), 742-749

you can explicitly find the following theorem:

*

*Given one PL structure on $M$, the isotopy classes of PL
structures on $M$ are in (1-1)-correspondence with the elements of
$H^3(M; \pi_3(TOP/PL))$.

